An Example of Bianchi Transformation in E⁴
We describe a particular class of pseudo-spherical surfaces in E⁴ which admit Bianchi transformations
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irk-123456789-1067212016-10-04T03:02:27Z An Example of Bianchi Transformation in E⁴ Gorkavyy, V. We describe a particular class of pseudo-spherical surfaces in E⁴ which admit Bianchi transformations Описан специальный класс псевдосферических поверхностей в E⁴, допускающих преобразования Бианки 2012 Article An Example of Bianchi Transformation in E⁴ / V. Gorkavyy // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 3. — С. 240-247. — Бібліогр.: 8 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106721 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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We describe a particular class of pseudo-spherical surfaces in E⁴ which admit Bianchi transformations |
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Gorkavyy, V. |
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Gorkavyy, V. An Example of Bianchi Transformation in E⁴ Журнал математической физики, анализа, геометрии |
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An Example of Bianchi Transformation in E⁴ |
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An Example of Bianchi Transformation in E⁴ |
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An Example of Bianchi Transformation in E⁴ |
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An Example of Bianchi Transformation in E⁴ |
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An Example of Bianchi Transformation in E⁴ |
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example of bianchi transformation in e⁴ |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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An Example of Bianchi Transformation in E⁴ / V. Gorkavyy // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 3. — С. 240-247. — Бібліогр.: 8 назв. — англ. |
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Журнал математической физики, анализа, геометрии |
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Journal of Mathematical Physics, Analysis, Geometry
2012, vol. 8, No. 3, pp. 240–247
An Example of Bianchi Transformation in E4
V. Gorkavyy
Mathematics Division, B. Verkin Institute for Low Temperature Physics and Engineering
National Academy of Sciences of Ukraine
47 Lenin Ave., Kharkiv 61103, Ukraine
E-mail: gorkaviy@ilt.kharkov.ua
Received March 2, 2011
We describe a particular class of pseudo-spherical surfaces in E4 which
admit Bianchi transformations.
Key words: pseudo-spherical surface, Bianchi transformation, Beltrami
surface.
Mathematics Subject Classification 2010: 53A07, 53B25.
Introduction
The aim of this note is to describe a particular class of two-dimensional
pseudo-spherical surfaces, which admit the Bianchi transformation, in the four-
dimensional Euclidean space.
Recall the classical definition of the Bianchi transformation, see [1, 2, 3]. Let
F be a pseudo-spherical surface, i.e. a surface of the constant negative Gauss
curvature K ≡ −k2, in the three-dimensional Euclidean space E3. Suppose
that F is represented in E3 by a position-vector r(ϕ, v) in terms of horocyclic
coordinates (ϕ, v), i.e. the metric form of F reads ds2 = 1
k2 dϕ2+e2ϕdv2. Consider
a new surface F ∗ whose position vector is
r∗ = r − ∂ϕr. (1)
It is well known that F ∗ is pseudo-spherical and has the same Gauss curva-
ture, K ≡ −k2; F ∗ is called a Bianchi transform of F . Using different horo-
cyclic coordinates and applying the Bianchi transformation, one can construct a
one-parameter family of various pseudo-spherical surfaces from a given pseudo-
spherical surface. Notice that the Bianchi transformation possesses some excep-
tional features in terms of geodesic congruences which may be used to suggest a
synthetic definition of the Bianchi transformation equivalent to (1).
c© V. Gorkavyy, 2012
An Example of Bianchi Transformation in E4
A direct generalization of the classical theory of Bianchi transformations to the
case of n-dimensional pseudo-spherical submanifolds in the (2n− 1)-dimensional
Euclidean space was suggested and described by Yu. Aminov in [4], see also
[1, 2, 5].
On the other hand, the question of how to extend the concept of the Bianchi
transformation to the case of n-dimensional pseudo-spherical submanifolds in N -
dimensional Euclidean spaces with arbitrary n ≥ 2, N ≥ 2n remains unsolved.
This open problem was supplied by Yu. Aminov and A. Sym in [6], and this is
just what motivated our results in this note.
In the simplest non-trivial case of n = 2, N = 4, if one asks to extend
the Bianchi transformation to the case of two-dimensional surfaces in the four-
dimensional Euclidean space E4, a reasonable way is to accept the same formula
(1) in order to construct a new surface F ∗ from a given pseudo-spherical surface
F ⊂ E4. Naturally, F ∗ is called a Bianchi transform of F provided that F ∗
is pseudo-spherical. However, it turns out that generically F ∗ is not pseudo-
spherical and thus a generic pseudo-spherical surface in E4 does not admit Bianchi
transforms [6].
Pseudo-spherical surfaces in E4 admitting Bianchi transforms were described
in [7] in terms of solutions of some particular system of partial differential equa-
tions GCR, which may be viewed as a generalization of the sine-Gordon equation.
The description deals with the fundamental forms of surfaces. However, no para-
metric representations for such particular pseudo-spherical surfaces were derived
and no one concrete example was presented. Our note is just aimed to remove
this gap.
First, in Sec. 1 we recall the classical construction of the Bianchi trasfor-
mation for pseudo-spherical surfaces in E3. Next, in Sec. 2 we describe a con-
structive method for producing a pseudo-spherical surface in E4 from a given
pseudo-spherical surface in E3, such surfaces in E4 will be referred to as stretched.
It is proved that an arbitrary stretched pseudo-spherical surface in E4 admits a
Bianchi transform and this Bianchi transform is stretched too. Relations between
the stretched pseudo-spherical surfaces in E4 and the solutions of the mentioned
GCR-system of [7] are analyzed in Sec. 3. As consequence, it is shown that there
exist pseudo-spherical surfaces in E4, which are not stretched but admit Bianchi
transforms (it should be quite interesting to find an explicit representation for
these surfaces). Finally, in Sec. 4 we describe the stretched pseudo-spherical sur-
faces in E4 produced from the standard pseudo-sphere (Beltrami surface) in E3.
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 241
V. Gorkavyy
1. Classical Theory of Bianchi Transformation
Let F̃ be a regular two-dimensional surface of the constant negative Gauss
curvature K ≡ −k2 in E3. Locally F̃ is parameterized by the horocyclic coor-
dinates (ϕ, v) so that its metric form reads ds̃2 = 1
k2 dϕ2 + e2ϕdv2. From the
intrinsic point of view, the coordinate curves v = const are parallel geodesics,
whereas ϕ = const are horocircles in F̃ .
Generically, given a horocyclic coordinate system (ϕ, v), one can locally pa-
rameterize F̃ by another local coordinate system (u, v) so that the coordinate
curves u = const and v = const form a conjugate net in F̃ . Then the metric form
reads
ds̃2 =
1
k2
dϕ(u, v)2 + e2ϕ(u,v)dv2, (2)
whereas the second fundamental form is diagonalized, b̃ = b̃11du2 + b̃22dv2. Ap-
plying the fundamental Codazzi equations, it is easy to show that
b̃11 = e−ϕ∂uϕ, b̃22 = −e3ϕ∂uϕ (3)
after some rescaling u → f(u). Moreover, the fundamental Gauss equation reads
∂uue2ϕ + ∂vve
−2ϕ + 2k2 = 0. (4)
Thus, generically any pseudo-spherical surface in E3 generates a solution of the
nonlinear pde (4). In its turn, due to the classical Bonnet theorem, any solu-
tion of (4) generates via (2), (3) a pseudo-spherical surface in E3 parameterized
by conjugate coordinates, whose one family of the coordinate curves is parallel
geodesics.
Let ρ(u, v) be the corresponding position vector of F̃ . Consider a new surface
F̃ ∗ in E3 represented by the position vector
ρ∗ = ρ− ∂ϕρ = ρ− 1
∂uϕ
∂uρ. (5)
It is easy to get that the metric form of F̃ ∗ reads ds̃∗2 = e−2ϕdu2 + 1
k2 dϕ2. Hence,
if ∂vϕ 6= 0, then the surface F̃ ∗ is regular and has the constant negative Gauss
curvature K = −k2. Thus, F̃ ∗ is pseudo-spherical and it is called a Bianchi
transform of F̃ .
The described Bianchi transformation of the pseudo-spherical surfaces in E3
has a number of remarkable geometric properties [3]. From the analytical point
of view, it corresponds to the involuting transformation ϕ(u, v) → ϕ∗(u, v) =
−ϕ(v, u) for the solutions of (4). Moreover, it may be interpreted as a particular
transformation for the solutions of the sin-Gordon equation.
242 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3
An Example of Bianchi Transformation in E4
2. Stretched Pseudo-Spherical Surfaces and Bianchi
Transformation
Now, view E3 as a horizontal hyperplane x4 = 0 in E4 and hence consider
the above surface F̃ as a surface in E4. Given F̃ ⊂ E3 ⊂ E4, define a new
two-dimensional surface F in E4 by
r(u, v) = (ρ(u, v), Aϕ(u, v) + B) , (6)
where A 6= 0, B are constant. Because of (2), the metric form of F is
ds2 = ds̃2 + A2dϕ2 =
(
A2 +
1
k2
)
dϕ2 + e2ϕ(u,v)dv2. (7)
It is easy to show that the Gauss curvature of F is K = − k√
A2k2 + 1
. Hence
F is pseudo-spherical, and the local coordinates (ϕ, v) in F are horocyclic. It
should be natural to say that the pseudo-spherical surface F ⊂ E4 is obtained by
stretching the pseudo-spherical surface F̃ ⊂ E3 ⊂ E4. Thereby F is referred to
as stretched, whereas F̃ is called the base of F . Evidently, the stretched pseudo-
spherical surfaces form a particular class of the pseudo-spherical surfaces in E4.
Let us apply to F the transformation
r∗ = r − ∂ϕr = r − 1
∂uϕ
∂ur. (8)
The vector function r∗ represents a new surface F ∗ in E4.
Proposition 1. F ∗ is a stretched pseudo-spherical surface. Moreover, the
base of F ∗ is the Bianchi transform F̃ ∗ of the base F̃ of F .
P r o o f. Due to (6), we have
r∗ =
(
ρ− 1
∂uϕ
∂uρ,Aϕ + B −A
)
. (9)
In view of (5), ρ∗ = ρ− 1
∂uϕ∂uρ represents exactly the Bianchi transform F̃ ∗ of F̃ .
Moreover, the metric form of F̃ ∗ is ds̃∗2 = e−2ϕdu2 + 1
k2 dϕ2, hence ϕ∗(u, v) =
−ϕ(v, u). Therefore, (9) may be rewritten as follows:
r∗ = (ρ∗, A∗ϕ∗ + B∗) , (10)
where A∗ = −A,B∗ = B −A. Comparing (6) with (10), one can easily conclude
that F ∗ is a stretched pseudo-spherical surface whose base surface is F̃ ∗. Notice
that the Gauss curvature of F ∗ is still the same, K = − k√
A2k2 + 1
, q.e.d.
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 243
V. Gorkavyy
Thus, any stretched pseudo-spherical surface in E4 admits a Bianchi trans-
form which is a stretched pseudo-spherical surface too. Besides, the Bianchi
transformation of the stretched pseudo-spherical surfaces in E4 is generated by
the classical Bianchi transformation of their base surfaces in E3.
R e m a r k. The same stretching procedure was applied in [8] to produce
two-dimensional pseudo-spherical surfaces, which admit Bianchi transforms, in
Riemannian products Mn × R1, where Mn is the sphere Sn or the Lobachevski
space Hn. It turns out that a pseudo-spherical surface in M3 × R1 admits a
Bianchi transform if and only if it is either a stretched surface or a hypersurface
in a horizontal slice M3 × {h0} ⊂ M3 × R1. As we will see in the next section,
this is not true for the case of R3 ×R1, i.e. if M3 = E3.
3. Stretched Pseudo-Spherical Surfaces and Solutions of the
GCR-System
Pseudo-spherical surfaces in E4 admitting Bianchi transforms were described
in [7]. Roughly speaking, a pseudo-spherical surface with K ≡ −1 in E4, which
is not a hypersurface in any hyperplane E3 ⊂ E4, admits a Bianchi transform
if and only if it can be parameterized in such a way that its fundamental forms
read
ds2 = dϕ2 + e2ϕdv2, (11)
II1 = e−ϕ∂uϕdu2 − e3ϕ∂uϕdv2, II2 = eϕPdv2, (12)
µ12 = Qdu, (13)
where the functions ϕ(u, v), P (u, v) and Q(u, v) satisfy the Gauss–Codazzi–Ricci
equations
∂uue2ϕ + ∂vve
−2ϕ + 2(PQ + 1) = 0, (14)
∂uP −Qe2ϕ∂uϕ = 0, (15)
∂vQ + Pe−2ϕ∂vϕ = 0 (16)
and the regularity conditions
∂uϕ 6= 0, ∂vϕ 6= 0, P 6= 0, Q 6= 0. (17)
Due to the classical Bonnet theorem, any solution {ϕ, P, Q} of the GCR-system
(14)–(17) generates a pseudo-spherical surface with K ≡ −1 in E4 which admits
a Bianchi transform.
Since the stretched pseudo-spherical surfaces in E4 admit Bianchi transforms,
they correspond to some particular solutions of (14)–(17).
244 Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3
An Example of Bianchi Transformation in E4
Proposition 2. The stretched pseudo-spherical surface F in E4, represented
by (6) with A =
√
k2−1
k , has the following fundamental forms:
ds2 = dϕ2 + e2ϕdv2, (18)
II1 = e−ϕ∂uϕdu2 − e3ϕ∂uϕdv2, II2 = e2ϕ
√
k2 − 1 dv2, (19)
µ12 = e−ϕ
√
k2 − 1 du. (20)
P r o o f. Set A =
√
k2−1
k in (6). Then (7) implies (18).
Differentiate (6) and write the vectors tangent to F
∂ur =
(
∂uρ,
√
k2 − 1
k
∂uϕ
)
, ∂vr =
(
∂vρ,
√
k2 − 1
k
∂vϕ
)
. (21)
The normal plane to F ⊂ E4 is spanned by the following unit vectors:
N1 = (n, 0) , N2 =
(
−
√
k2 − 1
∂uϕ
∂uρ,
1
k
)
, (22)
where n is the unit vector normal to the base surface F̃ ⊂ E3 ⊂ E4. Differentiate
(21) and find the second fundamental forms IIσ = 〈d2r,Nσ〉 of F with respect
to the normal frame (22); this yields (19).
Finally, differentiate (22) and find µ12 = 〈dN1, N2〉; this proves (20), q.e.d.
Comparing (18)–(20) with (11)–(13), we can see that the stretched pseudo-
spherical surface F corresponds to the solution
{
ϕ, P =
√
k2 − 1eϕ, Q =
√
k2 − 1e−ϕ
}
, (23)
whereas ϕ(u, v) is determined by the base F̃ and solves equation (11) which
reduces to (4). This solution was presented in the formula (32) of [7], where one
has to set c1 = 0, c2 =
√
k2 − 1.
Notice that (11), (12) has other solutions different from (23), see [7]. It means
that there are pseudo-spherical surfaces in E4 which are neither stretched surfaces
nor hypersurfaces in E3 ⊂ E4 but admit Bianchi transforms.
4. An Example: Stretched Pseudo-Spheres in E4
Let F̃ ⊂ E3 be a pseudo-sphere represented by the position vector
ρ(ϕ, v) = (eϕ cos v, eϕ sin v, Ψ) ,
Journal of Mathematical Physics, Analysis, Geometry, 2012, vol. 8, No. 3 245
V. Gorkavyy
where Ψ(ϕ) satisfies (Ψ′)2 + e2ϕ = 1
k2 , hence
Ψ = ±1
k
(√
1− k2e2ϕ +
1
2
ln(1−
√
1− k2e2ϕ)− 1
2
ln(1 +
√
1− k2e2ϕ)
)
. (24)
The local coordinates (ϕ, v) in F̃ are horospherical since ds̃2 = 1
k2 dϕ2 + e2ϕdv2.
However, if we apply the Bianchi transformation (1), then the transformed surface
F̃ ∗ degenerates to a curve (the axis of rotation of F̃ ). So we need some other
horocyclic coordinates in F . Such coordinates are given by
ϕ = − ln
(
2e−ϕ̂
k2v̂2 + e−2ϕ̂
)
, v =
2v̂
k2v̂2 + e−2ϕ̂
. (25)
In fact, it is easy to verify that the metric form of F̃ reads ds̃2 = 1
k2 dϕ̂2 + e2ϕ̂dv̂2,
so the local coordinates (ϕ̂, v̂) in F̃ are horocyclic.
Taking F̃ as the base, a stretched pseudo-spherical surface F in E4 is repre-
sented by the position vector
r(ϕ̂, v̂) = (eϕ cos v, eϕ sin v, Ψ, Av̂ + B) ,
where ϕ(ϕ̂, v̂), v(ϕ̂, v̂), Ψ(ϕ(ϕ̂, v̂)) are explicitly given by (24), (25), and A 6= 0, B
are arbitrary constants. In terms of the original coordinates (ϕ, v), the stretched
surface F is represented by
r(ϕ, v) =
(
eϕ cos v, eϕ sin v, Ψ(ϕ), A ln
(
1 + v2e2ϕk2
2
)
+ B
)
.
This surface in E4 should be called a stretched pseudo-sphere (a stretched
Beltrami surface). Applying the Bianchi transformation, one may obtain a new
sequence of the stretched pseudo-spherical surfaces in E4.
References
[1] Yu.A. Aminov, Geometry of Submanifolds. Gordon & Breach Science Publ., Ams-
terdam, 2001.
[2] K. Tenenblat, Transformations of Manifolds and Applications to Differential Equa-
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[3] Nonlinearity & Geometry. Luigi Bianchi Days. (D. Wojcik and J. Cieslinski, Eds.),
Polish Scientific Publishers PWN, Warsawa, 1998.
[4] Yu. Aminov, A Bianchi Transformation for a Domain of the Many-Dimensional
Lobachevski Space. — Ukr. Geom. Sb. 21 (1978), 3–5. (Russian)
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